material balance equation

TPG4150 Reservoir Recovery Techniques 2012 Material Balance Equations

Department of Petroleum Engineering and Applied Geophysics Professor Jon Kleppe Norwegian University of Science and Technology August 23, 2012

1

Material Balance Equations To illustrate the simplest possible model we can have for analysis of reservoir behavior, we will start with derivation of so-called Material Balance Equations. This type of model excludes fluid flow inside the reservoir, and considers fluid and rock expansion/compression effects only, in addition, of course, to fluid injection and production. First, let us define the symbols used in the material balance equations: Symbols used in material balance equations

Bg Formation volume factor for gas (res.vol./st.vol.)

Bo Formation volume factor for oil (res.vol./st.vol.) Bw Formation volume factor for water (res.vol./st.vol.) Cr Pore compressibility (pressure-1) Cw Water compressibility (pressure-1) ΔP P P2 1− Gi Cumulative gas injected (st.vol.) Gp Cumulative gas produced (st.vol.)

m Initial gas cap size (res.vol. of gas cap)/(res.vol. of oil zone) N Original oil in place (st.vol.) Np Cumulative oil produced (st.vol.)

P Pressure Rp Cumulative producing gas-oil ratio (st.vol./st.vol) = G Np p/ Rso Solution gas-oil ratio (st.vol. gas/st.vol. oil) Sg Gas saturation

So Oil saturation Sw Water saturation T Temperature Vb Bulk volume (res.vol.) Vp Pore volume (res.vol.)

We Cumulative aquifer influx (st.vol.) Wi Cumulative water injected (st.vol.) Wp Cumulative water produced (st.vol.) ρ Density (mass/vol.) φ Porosity

Then, the Black Oil fluid phase behavior is illustrated by the following figures: Fluid phase behavior parameters (Black Oil) Bo Rso

P P P P

BwB g



2

Oil density: ρρ ρ

ooS gS so

o

RB

=+

Water compressibility: CV

VPw

w

wT= −( )( )1 ∂

∂

Water volume change: B B e B 1 c Pw2 w 1c P

w1 ww= ≈ −− Δ Δ( )

Finally, we need to quantify the behavior of the pores during pressure change in the reservoir. The rock compressibility used in the following is the pore compressibility, and assumes that the bulk volume of the rock itself does not change. Pore volume behavior

Rock compressibility: CPr T= ( )( )1

φ∂φ∂

Porosity change: φ φ φw2 w1c P

w1 re 1 c Pr= ≈ +Δ Δ( ) The material balance equations are based on simple mass balances of the fluids in the reservoir, and may in words be formulated as follows: Principle of material conservation

Amount of fluids presentin the reservoir initially

(st. vol.)

Amount of fluids produced

(st. vol.)

Amount of fluids remainingin the reservoir finally

(st. vol.)

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪−⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪=⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

We will define our reservoir system in terms of a simple block diagram, with an initial reservoir stage before production/injection starts, and a final stage at which time we would like to determine pressure and/or production. Block diagram of reservoir

Gas

Oil

Water

Initial stage (1)

Gas

Oil

Water

Final stage (2)

oil production: Npgas production: RpNpwater production: Wp

aquifer influx: We

gas injection: Giwater injection: Wi



3

The two stages on the block diagram are reflected in the fluid phase behavior plots as follows: Initial and final fluid conditions

B

B

P

P

o Rso

P

wg

P

| |

|

||

|

|

|

1

11

1

2 2

22

B

Note: If a gas cap is present ini-tially, then the initial pressure isequal to the bubble point pressure

Now, we will apply the above material balance equation to the three fluids involved, oil, gas and water: Equation 1: Oil material balance

Oil presentin the reservoir

initially(st. vol. )

Oilproduced(st. vol. )

Oil remainingin the reservoir

finally(st. vol. )

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

−⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪=

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

or N - N = V S /Bp p o o2 2 2 yielding

SN N BVo

p o2

p22 =

−( )

Equation 2: Water material balance

Water presentin the reservoir

initially(st. vol. )

Waterproduced(st. vol. )

Waterinjected(st. vol. )

Aquiferinflux

(st. vol. )

Water remainingin the reservoir

finally(st. vol. )

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

−⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪=

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

or V S /B - W + W + W = V S /Bp w w p i e p w w1 1 1 2 2 2 yielding

( ) ( )S 1 m NBSS

1B

W W WBVw2 o1

w1

w1 w1i e p

w2

p2= +

−⎛⎝⎜

⎞⎠⎟⎛⎝⎜

⎞⎠⎟ + + −

⎡

⎣⎢⎢

⎤

⎦⎥⎥1



4

Equation 3: Gas material balance

Solution gaspresent in the

reservoir initially(st. vol.)

Free gas present in the

reservoir initially(st. vol. )

Gasproduced(st. vol. )

Gasinjected(st. vol. )

Solution gaspresent in the

reservoir finally(st. vol. )

Free gaspresent in the

reservoir finally(st. vol. )

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

+

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

−⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

=

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

+

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

or

�

NRso1 + mN(Bo1

Bg1) − NpRp + Gi = (N − Np )Rso2 + Sg2

Vp2

Bg2

yielding

�

Sg2 = N (Rso1 − Rso2) + m(Bo1

Bg1)

⎡

⎣ ⎢

⎤

⎦ ⎥ − Np (Rp − Rso2) + Gi

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪ (Bg2

Vp2)

In addition to these three fluid balances, we have the following relationships for fluid saturations and pore volume change: Equation 4: Sum of saturations

S S S 1.0o w g+ + =

Equation 5: Pore volume change

V =V ( +c P)p p r2 1 1 Δ



5

By combining the 5 equations above, and grouping terms, we obtain the material balance relationships, as shown below: THE COMPLETE BLACK OIL MATERIAL BALANCE EQUATION:

( ) ( )F N E mE E W W B G Bo g f w i e w2 i g 2= + + + + +, where production terms are

( )[ ]F N B R R B W Bp o2 p so2 g2 p w2= + − +

oil and solution gas expansion terms are

( ) ( )E B B R R Bo o2 o1 so1 so2 g2= − + − gas cap expansion terms are

E BBB

1g o1g2

g1= −

⎛

⎝⎜⎜

⎞

⎠⎟⎟

and rock and water compression/expansion terms are

( )E 1 m BC C S1 S

Pf w o1r w w1

w1, = − +

+−

Δ



6

MATERIAL BALANCE EQUATION FOR A CLOSED GAS RESERVOIR The material balance equation for a closed gas reservoir is very simple. Applying the mass balance principle to a closed reservoir with 100% gas, we may derive the general eguation

�

GBg1 = (G −Gp )Bg2 where

�

G is gas initially in place,

�

Gp is cumulative gas production, and

�

Bg is the formation-volume-factor for gas. Since

�

Bg is given by the real gas law

�

Bg = (constant) ZP

(here temperature is assumed to be constant)

the above material balance equation may be rewritten as

�

G Z1P1

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = (G −Gp )

Z2P2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

or

�

P2Z2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = (1−

Gp

G) P1Z1

⎛ ⎝ ⎜

⎞ ⎠ ⎟

This equation represents a straight line relationship on a

�

P2Z2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ vs.

�

Gp . plot. The line passes through

�

P1Z1

⎛ ⎝ ⎜

⎞ ⎠ ⎟ at

�

Gp = 0 , and through

�

G at

�

P1Z1

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = 0 . By making a best-fit straight line to measured data, and extrapolate, we

may get an estimate of

�

G .

The straight-line relationship is very useful in estimating the initial volume of gas-in-place (

�

G) from limited production history.

material balance equation

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